|
|
In this post, I will focus on the underappreciated topic of default correlation, how it might be related to equity correlation, and what it means for the VaR of your corporate credit portfolio. Balanced fund managers take note! To begin, let's establish some terminology. Consider obligors A and B, and a time horizon T. Let's denote the probability of default of A before T by pA and similarly the probability of default of B before T by pB.
Now for a little pop quiz.
Question: With knowledge of pA and pB you can determine which of the following?
- PAB The conditional probability that A defaults given B has defaulted
- PAB The joint probability that both A and B default
- rAB The linear correlation coefficient between the default indicator events IA and IB
- None of the above
Unfortunately, knowledge of the marginal probabilities of default is insufficient to determine the conditional probabilities of default, the joint probability, or the linear correlation. So the answer is none of the above. Knowledge of the marginal distributions and the linear correlation of default is, however, enough to determine the conditional default probabilities, and the joint probability, of default. The relationships between these quantities are given by:  Before discussing how one might go about estimating the linear correlation of default, let's just take a moment to see just how important this number is for credit risk. Let's suppose that the individual default probability for each obligor is 2%, so that pA = pB = 2%. Below we plot the joint probability of default, and the (identical) conditional probabilities of default.   It is important to note that the linear default correlation completely dominates both the conditional and joint probabilities of default. The joint probability of default, for example, is roughly 10 times as large under a correlation of 20% as under 0%, and the magnitude of this effect increases as the individual probabilities decrease. This is deferent from our experience with correlation as it applies to equity return variance, where the marginal effect of a lower average correlation is constant on a relative basis. Thus the relative value of proper diversification increases as the as risk decreases for credit sensitive portfolios.
Given that historical analysis by the ratings agencies suggests that the five-year cummulative default rates of the majority of investment grade rated bonds is below 2%, this suggests that for a typical investment grade portfolio of credits, default correlation should dominate the risk calculus. In particular, the use of average credit rating by some bond funds to summarize credit risk is inappropriate, not because it hides a few lower quality bonds, which may skew the loss risk, but because it provides absolutely no information about the dominant risk: the default correlation.
To illustrate the impact of default correlation on the portfolio VaR, consider the case of an equally weighted portfolio of 100 obligors with identical independent individual probabilities of default of p = 5% over a horizon T and zero recovery. Neglecting the interest rate component, the VaR over the horizon of this portfolio is characterized simply by the number of defaults over that horizon. In this example, we can compute the probability of there being k or fewer defaults using the cumulative Binomial distribution function Below we display the 99% VaR for this stylized portfolio.  As with equities, fully determining the joint probability of default would require us to estimate all the pair wise correlations within a portfolio. Even for a portfolio of 100 obligors, this would require the specification of 4,950 pair wise correlations, which would in turn require obtaining at a minimum 4,950 default events. This is simply not feasible. Even for the equity world, where there is ample return data, a factor structure is employed to make the correlation problem tractable.
The solution for corporate bonds is twofold. First, we employ a factor model structure on the equity returns to dimension reduce the problem. Second, we will apply a firm value or structural model of default (e.g., Merton Model) to link equity return correlation to asset return correlation and asset return correlation to default correlation.
To keep things simple for now, let's consider a one factor model of the firm assets directly (I will incorporate equity data in a later blog post). Specifically, let's assume that all assets are driven by a single, common factor with a standard normal distribution.
 Because the idiosyncratic components are assumed to be uncorrelated, the covariance between the assets will be r. If we normalize the asset values first, then we can view r as an asset correlation. For the structural model part, we will assume a simple barrier model. Namely, an obligor defaults if its asset value at the horizon T falls below some critical level K.
If we assume that all obligors in an equally weighted portfolio have the same barrier K, then uncorrelated idiosyncratic components means that conditional on a realization of F, the probability of having m or fewer defaults in a N obligor portfolio is given by the Binomial cumulative distribution function (4) above. Since the common factor has standard normal distribution, the VaR is given by further integrating the Binomial above against the Gaussian density function. Below are the plots of the 99% VaR for a 100 obligor portfolio for different choices of r, where the barrier K is set so that zero correlation corresponds to a marginal probability of default of 2%.
 We can immediately see two things from the figure. First, the VaR converges to the independent binomially distributed case as correlation goes to zero (compare to the prior graph). Second, a 2% individual probability of default with a 20% asset correlation has a 99% VaR of 14. We would get the same VaR by having a 7.5% individual probability of default with a 0% asset correlation. To put this in context, this is like saying that an average BBB rated portfolio would bear the same credit risk (as measured by 99% VaR) as a BB+ porftolio. Three guesses on which portfolio would have a higher yield.
Here, I highlighted the importance of default correlation and through a simple factor model, showed how asset correlation dominates the risk equation for a portfolio of investment grade credits. In my next post, I'll go into more detail as to how an equity factor model can be harnassed to estimate the credit risk, and how correlation "stress testing" can be leveraged by looking at some real world examples.
Subscribe by e-mail to receive new Taking Risk posts as they are published and follow @FactSet on Twitter.
In the eight years I have been discussing risk with our clients in the Asia Pacific region, the dominant question I continue to receive is typically phrased: “Are there really any differences in the risk model providers?” Historically this has led to a discussion on the differences in each of the risk model provider’s methodologies. Last year, my colleague Chris Ellis put together a series of blog entries (part 1, part 2, part 3) answering that exact question, which I have since pointed my clients to for answers to their queries. However, inevitably, a couple of days later, I hear back from them with a rebuttal along the lines of: “For the U.S. that makes sense, but what about Asia?”
So I will build off of my colleague’s previous analysis and see if the same results hold for the Asia Pacific region. My analysis is going to be considerably simpler than what Chris put together given the relative size of the market in the Asia Pacific region. Style indices are few and far between; i.e., there are not many Asia Pacific ex-Japan mid-cap growth funds to speak of. For the purposes of this blog entry, I will focus my attention on the three major markets in the region: Australia, Asia ex-Japan, and Japan, and I will perform the same analysis previously conducted using the Northfield, APT, and Axioma risk models. (At their request, Barra will no longer be included in comparisons of the different vendor risk models.)
Using the FactSet LionShares database, I retrieved fund constituents for the 30 largest funds in each market as of June 30, 2010. My initial focus is tracking error (active risk) using the MSCI All Country Asia, S&P ASX 300, and TOPIX as the benchmarks for the three respective markets (Asia ex-Japan, Australia, and Japan). For the Asia Pacific region, I used the respective global models for each of the three providers, and for the Japan and Australia analysis, I used the country-specific models for the analysis. I will refer to the three models simply as A, B, and C. The purpose of this analysis is not to suggest which model is the “best,” as this analysis is not attempting to answer that question and may mislead the reader to conclusions about the “best” concept. I am trying to assess only whether there are differences between the providers.
Let’s first take a look at the average tracking error:
It’s fairly easy to visually see that Model C is suggesting a lower tracking error in all three markets with Model B portending the higher tracking error for all three markets. Model A seems to sit squarely in the middle, although leaning more towards Model C, especially in Australia. But the key question is, are the differences we see here statistically significant?
I used the same statistical technique used in the previous blog entry: the Welch’s T-test. I will deem any value greater than two to be statistically significant.
This test further bolsters the conclusions we were able to see fairly easily from reviewing the average tracking errors. While there is a discernable difference between Models A and C, we can see the differences are not significant. However, the significance between Models B and C is consistent for all three regions.
This was a very simple exercise; I would most likely want to include more funds and a more robust set of portfolios and benchmarks before making any firm conclusions, but we can observe from this brief analysis that two of the three risk models I used in this exercise do appear similar while the third appears to suggest a consistently higher tracking error. This builds upon my colleague’s previous analysis that there are indeed differences between the different risk model providers.
Please share your thoughts in the comments.
Subscribe by e-mail to receive new Taking Risk posts as they are published and follow @FactSet on Twitter.
by guest bloggers Mike Joel, Quantitative Specialist, London, and Matthew Van Der Weide, Quantitative Specialist, Amsterdam
Now that the craze of the World Cup is over, vuvuzelas have silenced, and Paul the Octopus has retired from making predictions, it seems like the time to evaluate. Could it have been luck, or was Paul the Octopus really skilled, and, if so, where would that rank him against some of the greatest fund managers?
The first argument of sceptics is that there are potentially thousands, if not millions, of animals with assumed predictive skills and only those who predict correctly come to the surface. That does sound like a plausible story; had Paul been wrong in the group phase, surely we would not have heard as much of our squishy oracle. In a way this reminds us of the (e)mail scams that recommend a stock every week, building up a track record week after week, trying to persuade you to invest. Although these stories sound compelling if there are a number of "correct" predictions in a row, it cannot be ignored that these frauds simply stop writing the people for whom they made a wrong prediction. Something that Nassim Taleb has shown all to clear in his book Fooled by Randomness: in the end this is simply a numbers game. Eight out of eight predictions right and given 50/50 odds (we will address both assumptions later), that would mean that there is a 1 in 256 chance for a random animal in a German zoo becomes the next Oracle.
But that is not the complete story, Paul actually has a bit of a track record. Although it is doubtful that it improves his statistics, at least it shows he has been around and we can conduct some analysis. Back in 2008, he made predictions for the Euro 2008 Championship and got four out of six right then. Still assuming 50/50 odds, would you have invested with a manager that has Paul’s track record? A simple coin flip would have given you three out of six (assuming 50/50 odds, mutual exclusivity of events, and the caveat that you might need to quit your job to be able to repeat the experiment enough to obtain stable results).
So when we look at the total track record of the eight-legged oracle, he predicted 12 out of 14 matches correctly (again, of these mutually exclusive events!). Through the use of our friend, the factorial, we can calculate the probability of achieving this:
 where x = number of occurrences and y = number of wins
Which ultimately translates to 91/(2^14)=0.0055, or 0.55% probability.
So what about those odds? As we pointed out earlier, they are assumed to be 50/50. Ever wondered why sometimes the bets at a bookmaker do not seem to add up? Effectively for the group phase there is a one in three chance; teams can draw and bookies can make money. Hence, given the extra choice, the odds of making the right prediction on 12 out of 14 matches plummets to 0.0019%, which is still slightly better than getting five out of six numbers (not including the bonus number) correct in the National Lottery (~0.0018%)= £1,500. Although when placing a bet with a bookie, one (usually) has detailed knowledge of team, hence there is more than luck in play and the probabilities are probably biased.
So how does Paul hold up against a good or even average fund manager? Grinold and Kahn define the Information Ratio (IR) as the Information Coefficient (IC) times the Square Root of Opportunity
( ). If the IC is our measure of skill, we will assign this the probability of Paul correctly predicting 12 out of the 14 matches (for lack of a better measure). This gives us an IR of 12/14 * Sqrt(14) = 3.2, which is remarkable, although given the light track record, not reliable. To maintain the same IR going forward, Paul does not have to maintain his skill, but merely place more bets!
Now to compare him to a fund manager, how can we describe Paul the Octopus’ investment style? He is not your typical long-only equity investor; his style more resembles that of a hedge fund: long team A, short team B (given that one of the co-authors is Dutch we will not disclose the teams). If one has to put a label on what the World Cup is, "Event Driven" probably fits best. Looking at the information ratio over the last 14 months, for the Lipper Tass Event Driven hedge fund universe, where would that put Paul the Octopus? His performance is not too bad as he ranks comfortably amongst the best names in the top quartile.
This brings us back to the main question: is Paul’s ability the result of skill or luck (putting aside any bias that may influence the numbers, such as shape or colour of the flag, an inclination to favour Germany, etc)? Given Paul’s predictions and where he falls relative to other funds of his style, he is a highly skilled manager. But whether this is true skill, whether there is a correlation between the winning team and characteristics that attract Paul or whether this is pure luck, the jury is still out; there are just not enough data points to quantify skill vs. luck. Perhaps Paul’s situation is a case of survivorship bias to the extreme (also quite literal), but one thing is certain: the next time one peruses a marketing brochure, it may be worth contemplating Paul and his ability. Any number of calculations/statistics will make Paul look like one of best managers, and although the numbers may look good, one must delve into the details behind the numbers. The risks for managers involved in pursuing the stats and making it to a top quartile ranking, though, are minimal compared to being part of our next dish.
Subscribe by e-mail to receive new Taking Risk posts as they are published and follow @FactSet on Twitter.
|
|
Posts
